An efficient method for solving a correlated multi-item inventory system

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by  Chang-Yong Lee

An efficient method for solving a correlated multi-item inventory system
Chang-Yong Lee∗ and Dongju Lee
The Department of Industrial & Systems Engineering,
Kongju National University, Kongju 314-701 South Korea
(Dated: January 20, 2014)
Weproposeanefficientmethodoffindinganoptimalsolution foramulti-itemcontinuousreview
inventory model in which a bivariate Gaussian probability distribution represents a correlation
between the demands of different items. By utilizing appropriate normalizations of the demands,
4 weshow that thenormalized demandsare uncorrelated. Furthermore, theset of equationscoupled
1 with different items can be decoupled in such a way that the order quantity and reorder point
0 for each item can be evaluated independently from those of the other. As a result, in contrast
2 to conventional methods, the solution procedure for the proposed method can be much simpler
and more accurate without any approximation. To demonstrate the advantage of the proposed
n
a method,wepresentasolution schemefor amulti-itemcontinuousreviewinventorymodelinwhich
J thedemandof optionalcomponentsdependonthat ofa“vanilla box,”representingthecustomer’s
stochasticdemand,understochasticpaymentandbudgetconstraints. Wealsoperformasensitivity
7
1 analysis to investigate the dependence of order quantities and reorder points on the correlation
coefficient.

]
h
p
-
p I. INTRODUCTION
m
o In a competitive global market, a manufacturer usually needs to provide a wide variety of products with a short
c
amount of lead time to improve customer satisfaction and increase market share. When an order from a customer
.
s arrives,differentiationtothetailoredorderisoftenpostponeddowntheassemblylinetoachievebothproductvariety
c
and short lead time. In general, the more diverse a product, the longer lead time it requires. Modularization and
i
s postponement (or delayed differentiation) can be an effective means to reduce the lead time while maintaining a
y
wide variety of products. Many modularization and postponement studies have shown that these concepts offer an
h
p advantage in terms of reducing uncertainty and forecasting errors with regard to demand [8, 13, 14] in addition to
[ creating product variety and customization at low cost [5]. Thus, modularization and postponement have become

importantconceptsinthemarkettoprovidebetterservicetocustomersandmakethebusinessprocessmoreefficient.
1
Inaproductlinesuchascomputerretailingorautomobileassembly,conceptsofmodularizationandpostponement
v
are realized by an assembly process that consists of a semi-finished product “vanilla box” and optional components
7
1 that are directly used in the final assembly. The vanilla box consists of components, known as the commonality of
3 parts, needed to assemble the final product with appropriate optional components, and the vanilla box approachhas
4 been shown to be effective under high variance [13].
1. Applications of modularization and postponement to an inventory system require a multi-item model in which the
0 demand of each of the several optional components depends on the demand or the presence of the vanilla box. The
4 vanillaboxrepresentsthe customer’sstochasticdemand,andoptionalcomponents,inturn,dependonthe demandof
1 the vanilla box for final assembly. Thus, the demands of the vanilla box and optional components are stochastically
:
v correlated. As the inventory model becomes complicated because of the correlation,it is desirable to find an efficient
i and accurate method to solve the model system.
X
A multi-item inventory model was proposedto comply with the concept of modularizationand postponement [15].
r The modelconsists ofa vanilla box andoptionalcomponents inwhich the correlationbetweenthe two types ofitems
a
is implemented as a bivariate Gaussian probability distribution whereas the optional components are independent
of each other. This model handles a continuous review inventory system in which an order quantity Q is placed
wheneveraninventorylevelreachesacertainreorderpointrunderthepresenceofservicelevelandbudgetconstraints.
Subsequently, a stochastic paymentis also included in the model in such a manner that the total inventorycost does
not exceed a predetermined budget [16].
In this paper, we propose an efficient method to solve a correlated multi-item continuous review inventory model
in which the correlation between the vanilla box and an optional component is represented by a bivariate Gaussian
probability distribution. By using appropriate normalizations of the demands of items, we show that the set of
∗Electronicaddress: [email protected]
2
equations coupled with the vanilla box and optional components can be reduced to sets of decoupled equations
for each item. Furthermore, each set of decoupled equations is simplified in a closed form and solved without any
approximation. Thus, the equations for each item can be solved independently of each other.
The conventionalmethod for solving such a model system is based on a heuristic of combining a Newton–Raphson
method and a Hadley–Whitin iterative procedure [16]. At each iteration, a candidate solution is found by using the
Newton-Raphsonmethodinwhichnumericalintegrationsarecarriedoutwhererequired. Theiterationproceedsuntil
both Q and r sufficiently converge. Briefly, the conventional method takes the set of simultaneous equations for Q’s
andr’s asawholeandutilizesheuristicapproximatingprocedures. Giventhattheconventionalmethodusesarather
complicated approximation and iteration, it requires heavy computation time.
In contrast, the proposed method does not rely on any approximation or heuristics. As a result, the solution
procedure for the proposed method is much simpler, more accurate, and offers shorter computing time than the
conventional method. We apply the proposed method to a correlated multi-item continuous review inventory model
todemonstrateitsusefulness. Wealsoperformasensitivityanalysisintermsofthecorrelationtofurthercharacterize
thebehavioroftheorderquantityandreorderpointofoptionalcomponents. Inaddition,theproposedschemecanbe
used as a dependable method for a more generalized multi-item continuous review inventory model with much more
complicated correlations among items.
The restofthis paperis organizedasfollows. SectionII discussesthe normalizationofthe demandsandintroduces
an illustrative model to show how the set of simultaneous equations can be decoupled and simplified. Section III
describes the proposed method for solving the model system. Section IV presents experimental results and discusses
the sensitivity analysis. Section V summarizes the study and gives our conclusions.
II. NORMALIZATION AND MULTI-ITEM INVENTORY MODEL
A. Correlation and normalization
Consider a multi-item inventorymodel that includes the correlationbetween a vanilla box and anoptional compo-
nent. Furthermore, we allow multiple optional components and each optional component is dependent on the vanilla
box through a bivariate Gaussian probability distribution, whereas optional components are independent of each
other.
A bivariate Gaussian probability distribution function (PDF) of the random variables X and X of the demand
v j
for the vanilla box and the jth optional component, respectively, is given by
1 1
f(x ,x )= exp (~x ~µ)TΣ−1(~x ~µ) . (1)
v j
2π Σ −2 − −
| | (cid:26) (cid:27)
Here, the variable vector ~x, the mean vector ~µp, the covariance matrix Σ, and the correlation coefficient ρ between
j
X and X are expressed, respectively, as
v j
~x= xv , ~µ= µv , Σ= σv2 σvj , and ρ σvj (2)
(cid:20) xj (cid:21) (cid:20) µj (cid:21) (cid:20)σvj σj2 (cid:21) j ≡ σvσj
with Σ being the determinantofthe 2 2matrixofΣ. GiventhatX depends onX ,weexpressthebivariatePDF
j v
| | ×
as a product of the marginal PDF of X and the conditional PDF of X given X = x . In this way, the bivariate
v j v v
PDF of Eq. (1) can be written as
2
f(x ,x ) = 1 exp (xv−µv)2 1 exp xj − µj +ρjσσvj(xv −µv)
v j √2πσv (cid:26)− 2σv2 (cid:27)√2πσj 1−ρ2j −h (cid:16) 2σj2(1−ρ2j) (cid:17)i 
q
f (x ) f (x x ) . (3)
≡ Xv v Xj|Xv j| v  
Ingeneral,thedemandforthevanillaboxisequaltothecustomer’sdemand,whereasthedemandforeachoptional
component depends on the safety stock of the vanilla box. Given that the safety stock of the vanilla box is r µ ,
v v
−
the demand for each optional component depends on the reorder point of the vanilla box r . This implies that the
v
conditional PDF of X is evaluated at X = r . Motivated by this characteristic, we define the normalized random
j v v
variables as
X µ X µ
v v j oj
Z − and Z − , (4)
v j
≡ σ ≡ σ
v oj
3
where
σ
σ σ 1 ρ2 and µ µ +ρ j(r µ ) . (5)
oj ≡ j − j oj ≡ j jσ v− v
v
q
Note that the random variable Z contains not only the demand X of the jth optional component but also the
j j
reorder point r of the vanilla box. With the normalization, the bivariate PDF at X =r can be rewritten as
v v v
fXv(xv) fXj|Xv(xj|rv)= √12πe−zv2/2√12πe−zj2/2 . (6)
We see from Eq. (6) that the normalization decomposes the bivariate PDF into a product of two PDFs of Z and
v
Z , both of which can be regarded as independent standard univariate PDFs. This implies that the notion of the
j
conditional PDF does not exist and there is no distinction between dependent and independent items. In what
follows, we take a correlated multi-item continuous review inventory system as an illustration of the advantage of
normalization.
B. Model formulation and set of equations
The illustrative modelwe considerinthis paperis a correlatedmulti-item continuousreview inventorysystemthat
includes two types of items: a vanilla box and many optional components. Furthermore, as stated in Section IIA,
each optional component depends on the vanilla box through a bivariate Gaussian probability distribution. We use
the following notations to formulate the model:
A : fixed procuring cost,
•
C : unit variable procurement cost,
•
D : expected annual demand,
•
h : carrying cost,
•
p : unit shortage cost,
•
κ : service cost rate, and
•
β : available budget limit.
•
Notethateachofthesetermscanbeusedforboththe vanillaboxandoptionalcomponentswheneverpossible. We
distinguishthevanillaboxfromthejthoptionalcomponentbythesubscriptsv andoj,respectively. Forinstance,A
v
and A represent the fixed procuring costs of the vanilla component and the jth optional component, respectively.
oj
The model is composed of the sum of the expected average annual cost (EAC) of the two types of items under
budgetary constraint. The budgetary constraint, in turn, includes the service costs of the vanilla box and optional
components. A detailed account of the model development can be found in [15, 16].
The objective of the model is to minimize the sum of EAC for the vanilla box and m optional components under a
stochasticbudgetaryconstraint. Thatis,wewouldliketofind(Qv,rv)and(Q~o,r~o),whereQ~o =(Qo1,Qo2, ,Qom)
···
and r~o =(ro1,ro2, ,rom), that minimize
···
EAC(Q~,~r)=EAC1(Qv,rv)+EAC2(Q~o,r~o) (7)
subject to
m m
Prob C (Q +r X )+ C (Q +r X )+κ F (r )+ κ F (r r ) β η (8)
 v v v − v oj oj oj − j v Xv v oj Xj|Xv oj| v ≤ ≥
 Xj=1 Xj=1 
for  
Q 0, r 0 and Q 0, r 0 for j =1,2, ,m . (9)
v v oj oj
≥ ≥ ≥ ≥ ···
4
In addition, F (r ) and F (r r ) are the cumulative density functions (CDFs) of f (x ) and f (x x ),
Xv v Xj|Xv oj| v Xv v Xj|Xv j| v
respectively. EAC1 and EAC2 are given as
A D Q p D ∞
v v v v v
EAC1(Qv,rv) = Q +CvDv+hv 2 +rv−µv + Q (x−rv)fXv(x)dx , (10)
v (cid:18) (cid:19) v Zrv
m A D Q p D ∞
EAC2(Q~o,r~o) = j=1" oQjojoj +CojDoj +hoj(cid:18) 2oj +roj −µoj(cid:19)+ oQjojoj Zroj(x−roj)fXj|Xv(x|rv)dx#,(11)
X
where µ is defined as in Eq. (5). It is readily shown that the constraint of Eq. (8) can be rewritten as
oj
m m
Cv(Qv+rv)+ Coj(Qoj +roj)+κvFXv(rv)+ κojFXj|Xv(roj|rv)≤β+µY +z1−ησY , (12)
j=1 j=1
X X
where
m m
µ C µ + C µ , σ2 C2σ2+ C2 σ2 , (13)
Y ≡ v v oj oj Y ≡ v v oj oj
j=1 j=1
X X
and z1−η =F−1(1 η) with F−1(1 η) being the inverse of the standard Gaussian CDF of the probability η.
− −
With the model, the Lagrangianfunction J using the Lagrangianrelaxation can be written as
m
J = EAC1(Qv,rv)+EAC2(Q~o,r~o)+λCv(Qv+rv)+ Coj(Qoj +roj)
 Xj=1
m

+κvFXv(rv)+ κojFXj|Xv(roj|rv)−(β+µY +z1−ησY) . (14)
Xj=1 
The first order necessary conditions can be achieved by differentiating J with respect to Qv, rv, Qoj, roj, and λ:
∂J A D h p D ∞
oj oj oj oj oj
= + (x r )f (xr )dx+λC =0 (15)
∂Q − Q2 2 − Q2 − oj Xj|Xv | v oj
oj oj oj Zroj
∂J p D ∂ ∞ ∂
oj oj
= h + (x r )f (xr )dx+λC +λκ F (r r )=0 (16)
∂r oj Q ∂r − oj Xj|Xv | v oj oj∂r Xj|Xv oj| v
oj oj oj Zroj oj
∂J A D h p D ∞
v v v v v
= + (x r )f (x)dx+λC =0 (17)
∂Q − Q2 2 − Q2 − v Xv v
v v v Zrv
∂J p D ∂ ∞ m p D ∞ ∂
v v oj oj
= h + (x r )f (x)dx+ (x r ) f (xr )dx
∂r v Q ∂r − v Xv Q − oj ∂r Xj|Xv | v
v v v Zrv j=1 oj Zroj v
X
m m
σ d ∂
j
(h +λC ) ρ +λC +λκ F (r )+ λκ F (r r )=0 (18)
− oj oj jσ v vdr Xv v oj∂r Xj|Xv oj| v
j=1 (cid:18) v(cid:19) v j=1 v
X X
m m
∂J
∂λ = Cv(Qv+rv)+ [Coj(Qoj +roj)]+κvFXv(rv)+ κojFXj|Xv(roj|rv)−(β+µY +z1−ησY)=0.(19)
j=1 j=1
X X
Note that Eqs. (15)–(18) are simultaneous equations for Q ,r ,Q , and r .
oj oj v v
C. Simplification of equations using normalization
Given that Q and r are coupled with Q and r from Eqs. (15)–(18), the equations are intractable to solve
oj oj v v
directly. For example, two equations [Eqs. (15) and (16)] contain three variables Q , r , and r . It turns out,
oj oj v
however,that the normalizationsdiscussed in Section IIA cannot only simplify the various expressionsin Eqs.(15)–
(18), but also, more importantly, decouple the equations for the optional components [Eqs. (15) and (16)] from the
equations for the vanilla box [Eqs. (17) and (18)].
5
FIG. 1: The list of expressions that are used to simplify the first order necessary condition.
Similarly to the normalization of Eq. (4), we further define the normalized reorder points as
z rv−µv and z roj −µoj = (roj −µj)−ρjσσvj(rv −µv) , (20)
v oj
≡ σv ≡ σoj σ 1 ρ2
j − j
q
where we have used the definition of µ of Eq. (5). Note that the normalized reorder point z is a function of r
oj oj oj
and r .
v
With the normalization defined in Eqs. (4) and (20), the various expressions in Eqs. (15)–(18) can be simplified as
shown in Fig. 1. We derive the simplified expressions in detail in VI. With the normalization, Eqs. (15)–(19) can be
re-expressed as
∂J A D h p D
oj oj oj oj oj
= + σ L(z )+λC =0 (21)
∂Q − Q2 2 − Q2 oj oj oj
oj oj oj
∂J p D κ
oj oj oj
= h G(z )+λC +λ f(z )=0 (22)
oj oj oj oj
∂r − Q σ
oj oj oj
∂J A D h p D
v v v v v
= + σ L(z )+λC =0 (23)
∂Q − Q2 2 − Q2 v v v
v v v
∂J p D κ
v v v
= h G(z )+λC +λ f(z )=0 (24)
v v v v
∂r − Q σ
v v v
m m
∂J
= C (Q +σ z )+ C (Q +σ z )+κ F(z )+ κ F(z )
v v v v oj oj oj oj v v oj oj
∂λ
j=0 j=0
X X
(β+µY +z1−ησY)=0 . (25)
−
Here, we define
f(z) 1 e−z2/2 , G(z) ∞f(t) dt , F(z) 1 G(z) , and L(z) f(z) zG(z) . (26)
≡ √2π ≡ ≡ − ≡ −
Zz
Note that, owing to the normalization, Eqs. (15)–(18) are decoupled into two sets of equations: Eqs. (21) and
(22) and Eqs. (23) and (24). Furthermore, each set of equations is identical and differs from the other only by the
subscript. ThisimpliesthatEqs.(21)and(22)canbesolvedindependentlyfromEqs.(23)and(24). Thisisexpected
because the PDF of the bivariate Gaussian distribution [Eq. (3)] can be expressed as the product of the PDF of the
normalized variables Z and Z [Eq. (6)]. Thus, Eqs. (21) and (22) can be solved for Q and z directly. Similarly,
v j oj oj
Eqs. (23) and (24) can be solved for Q and z . Once z and z are found, we can use Eqs. (4) and (20) to get r
v v oj v oj
and r . In the next section, we describe how to solve the set of Eqs. (21)–(24) under the constraint of Eq. (25).
v
III. PROPOSED METHOD TO SOLVE THE MODEL SYSTEM
Similar to the approach by [9], the proposed method to solve the model system consists of two parts. First, we
regard Eqs. (21)–(24) as a subproblem for a given λ. Second, we repeatedly solve the subproblem until we find the
solution λ to Eq. (25).
6
A. Procedures for solving the subproblem
Foragivenλ,weregardEqs.(21)–(24)asasubproblemandsolvethemforQ ,z ,Q ,andz . Thecrucialpointis
oj oj v v
thatsolvingEqs.(21)and(22)forz andQ isindependentofsolvingEqs.(23)and(24)forz andQ ,respectively.
oj oj v v
Giventhat G(z ) canbe numericallyevaluated for anyz , we note that Eqs.(21) and(22) arefunctions ofQ and
oj oj oj
z only. Byeliminating Q fromEqs.(21)and(22), weobtainanequationforz . Inparticular,Eqs.(21)and(22)
oj oj oj
can be rewritten, respectively, in terms of Q as
oj
A D +p D σ L(z ) p D G(z )
Q2 = oj oj oj oj oj oj and Q = oj oj oj . (27)
oj h /2+λC oj h +λC +λ(κ /σ )f(z )
oj oj oj oj oj oj oj
We can eliminate Q from Eq. (27), resulting in an equation for g (z ) in terms of z only:
oj oj oj oj
1/2
p D G(z ) A D +p D σ L(z )
oj oj oj oj oj oj oj oj oj
g (z ) =0 . (28)
oj oj
≡ h +λC +λ(κ /σ )f(z ) − h /2+λC
oj oj oj oj oj (cid:26) oj oj (cid:27)
There exists a unique solution z =z∗ of Eq. (28) if the following three conditions are satisfied:
oj oj
(a) g (z ) is a continuous function in z .
oj oj oj
(b) g (z ) is strictly monotonic in z .
oj oj oj
(c) There exist two distinct values z1 and z2 such that goj(z1) goj(z2)<0.
Condition(a)isimmediatelysatisfiedbecausef(z ),G(z ),andL(z )arecontinuousinz . Thefollowingtheorem
oj oj oj oj
satisfies condition (b).
Theorem 1. g (z ) is strictly decreasing function in z .
oj oj oj
The proof is given in VII. Finally, condition (c) imposes a restriction on the values of the parameters. Given that
g (z ) is a continuous and strictly decreasing function in z from conditions (a) and (b), by assuming that z >0,
oj oj oj oj
the following two inequalities shouldbe satisfiedto meet condition (c): g (0)>0 and g (+ )<0. It is easy to see
oj oj
∞
that the condition g (+ ) <0 is satisfied by noting that G(+ )= 0 and 0 <L(+ )< . Therefore, the values
oj
∞ ∞ ∞ ∞
of the parameters have to satisfy g (0)>0. That is,
oj
1/2
1 p D A D +p D σ /√2π
oj oj oj oj oj oj oj
g (0)= >0 . (29)
oj
2hoj +λCoj +λκoj/(√2πσoj) −( hoj/2+λCoj )
The values of the parameters should satisfy this inequality for Eq. (28) to have a a unique solution.
Given that Eq. (28) is a function of z only, one can use a simple search technique, such as a bisection method
oj
(see, for instance, [[11]]), to solve it at least numerically if not analytically. Once the solution z∗ of g (z∗ ) = 0 is
oj oj oj
obtained, we can substitute it back into Eq. (27) to get Q∗ , the solution of Q . We repeat the same procedure for
oj oj
j =1,2, ,m to obtain the normalized reorder point and order quantity for the optional components.
···
The normalized reorder point z and order quantity Q for the vanilla box can be obtained by applying a method
v v
similar to that used for the optional components. That is, Eqs. (23) and (24) can be rewritten respectively in terms
of Q as
oj
A D +p D σ L(z ) p D G(z )
Q2 = v v v v v v and Q = v v v . (30)
v h /2+λC v h +λC +λ(κ /σ )f(z )
v v v v v v v
The existence of a unique solution z∗ of
v
p D G(z ) A D +p D σ L(z ) 1/2
v v v v v v v v v
g (z ) =0 (31)
v v
≡ h +λC +λ(κ /σ )f(z ) − h /2+λC
v v v v v (cid:26) v v (cid:27)
can be proven in a similar fashion to the case for z∗ of Eq. (28). Thus, we can solve numerically for z = z∗;
oj v v
subsequently, we can solve for Q =Q∗ by substituting z∗ into either Eq. (23) or (24).
v v v
7
TABLE I:Parameters for thevanilla box and thetwo optional components with η=0.9031 so that z1−η =−1.3.
A C D h p κ µ σ β
Vanilla box v v v v v v v v
700 150 10,000 6 8 4000 300 40 150,000
Optional component Aoj Coj Doj hoj poj κoj µj σj ρj
1 40 3 4000 0.7 1.0 200 100 15 0.5
2 20 2 6000 0.4 0.7 150 170 20 0.8
B. Algorithm for solving the model system
Thesolutionschemeforthe subproblemdiscussedinSectionIIIAreducesthesetofEqs.(21)–(25)tooneequation
[Eq. (32)] with one unknown λ, which is implicitly dependent on the variables:
m m
g(λ)=Cv(Qv+σvzv)+ Coj(Qoj +σojzoj)+κvF(zv)+ κojF(zoj) (β+µY +z1−ησY) . (32)
−
j=0 j=0
X X
If g(λ) > 0, then the constraint is violated; otherwise (that is, g(λ) 0), the constraint is satisfied. By using the
≤
Lagrangian relaxation [9], the proposed algorithm to obtain the optimal reorder points and order quantities for the
optional components and vanilla box [r , Q , r , Q ] is as follows:
oj oj v v
Step 1: Find λ1 and λ2 such that g(λ1)>0 and g(λ2)<0.
Step 2: For each λ1 and λ2, solve the subproblem as follows:
Step 2(a): For j =1 to m, numerically solve Eq. (28) for z to obtain z∗ , and substitute z∗ into Eq. (27) to
oj oj oj
get Q∗ .
oj
Step 2(b): Numerically solve Eq. (31) for z to obtain z∗, and substitute z∗ into Eq. (30) to get Q∗.
v v v v
Step 3: Let λnew = (λ1 +λ2)/2 and find Q∗oj, zo∗j, Q∗v, and zv∗ from Steps 2(a) and 2(b). If g(λnew) > 0, then let
λ1 =λnew; otherwise let λ2 =λnew.
Step 4: Repeat Steps 2 and 3 until g(λ1) <ǫ or g(λ2) <ǫ, where ǫ is a predetermined error.
| | | |
Step 5: Use Eq. (20) to get r∗ and r∗ from z∗ and z∗ , respectively.
v oj v oj
IV. EXPERIMENTAL RESULTS AND DISCUSSION
We illustrate the performance of the proposed method by an experiment that consists of one vanilla box and two
optional components (i.e., m = 2). The parameters for the vanilla box and two optional components are listed in
Table I. They are the same as the parameters used in [16] and ǫ=10−7. Table II lists the solution to the model for
a given input from Table I together with the results of [16].
We also perform a sensitivity analysis of the order quantities and reorder points for the optional components with
respecttothecorrelationcoefficient. ItshouldbenotedthatQ andr areindependentofρ fromEqs.(30)and(31).
v v j
For the sensitivity analysis, we first need to find the behavior of z as ρ varies. Figure 2 shows that z is almost
oj j oj
constant with respect to ρj although zo1 decreases slightly for large value of ρ1. This implies that the normalized
reorder points are insensitive to the correlation coefficient.
Figure 3 shows the behavior of order quantities Q of the optional components as the correlation coefficient ρ
oj j
varies. From the first equation in Eq. (27), we see that the dependence of Q on ρ stems from σ L(z ). Because
oj j oj oj
z is more or less insensitive to ρ [Fig. 2], so is L(z ). Thus, considering ρ dependence only, we have
oj j oj j
Q σ σ 1 ρ2 . (33)
oj ∝ oj ≈ j − j
q
This implies that Q decreases as the absolute value of ρ increases and Q reaches its maximum when ρ = 0 as
oj j oj j
shown in Fig. 3. The maximum of Q at ρ = 0 can also be proved as follows. From the first equation in Eq. (27),
oj j
it can be readily shown that
dQ p D L(z )+z G(z ) σ2
oj oj oj oj oj oj j
= ρ . (34)
j
dρ − h +2λC Q σ
j { oj oj}(cid:26) oj (cid:27) oj
8
TABLE II: Solutions for Qv, rv, Qoj, and roj, together with λ and thetotal cost.
Qv rv Qo1 ro1 Qo2 ro2 λ EAV(Q~,~r)
Proposed method 860.8246 341.6691 580.8890 121.5989 648.4425 202.7676 0.045190 1,536,070
[16] 862.3301 340.3125 579.6005 122.7817 647.4532 203.3750 0.045044 1,536,061
1.6
optional 1
1.5 optional 2
1.4
1.3
zoj1.2
1.1
1.0
0.9
0.8
0.0 0.2 0.4 ρ 0.6 0.8 1.0
j
FIG. 2: Plots of thenormalized reorder points zoj for theoptional componentsversus thecorrelation coefficient ρj.
Thus, Q is extreme when ρ =0. Furthermore,
oj j
d2Q p D L(z )+z G(z )
oj oj oj oj oj oj
= σ <0 (35)
dρ2j (cid:12)(cid:12)ρj=0 −{hoj +2λCoj}(cid:26) Qoj (cid:27) j
(cid:12)
(cid:12)
implies that Qoj has its maximum(cid:12)when ρj =0.
Figure4showsthebehaviorofreorderpointsr ofthe optionalcomponentsasthecorrelationcoefficientρ varies.
oj j
Unlike the order quantity, r reaches its maximum at a positive value of ρ . For the behavior of r , we can rewrite
oj oj oj
Eq. (20) as follows:
σ
r =µ +ρ j (r µ )+σ 1 ρ2 z . (36)
oj j jσ v − v j − j oj
v
q
Because z is almost independent of ρ , as ρ increases, the second term on the right-hand side of Eq. (36) also
oj j j
increases while the third term decreases. Thus, there is a trade-off between the second and the third terms, resulting
in an optimum value of r . Furthermore, the maximum r occurs when
oj oj
r µ
ρmax v− v . (37)
j ≈ σ2 z2 +σ2 (r µ )2
v oj j v− v
q
Thisimpliesthatρmax dependsonthesafetystockr µ ofthe vanillabox. Giventhatthesafetystockisapositive
quantity, we have 0j<ρmax <1 unlike the maximumv−of Qv .
j oj
V. SUMMARY AND CONCLUSION
In this paper, we presented an efficient method for finding an optimal solution (Q, r) to a correlated multi-item
continuous review inventory model in which a bivariate Gaussian probability distribution is used as a correlation
9
optional 1
optional 2
650
oj
Q
580
575
0.0 0.2 0.4 ρ 0.6 0.8 1.0
j
FIG. 3: Plots of theorder quantities Qoj for the optional components versus thecorrelation coefficient ρj.
122
120
1 206
o
r 204
118 202
ro2 200
198
196
116 ρ
0.0 0.3 0.6 0.9
2
ρ
0.0 0.3 0.6 0.9
1
FIG. 4: Plots of the reorder points ro1 for optional component 1 versus the correlation coefficient ρ1 while ρ2 = 0.5. Inset:
Same plot for optional component 2.
between the vanilla box and an optional component. By normalizations of the randomvariables for the demands, we
showed that the bivariate Gaussian PDF can be expressed as a product of two independent Gaussian PDFs, which
implies that the normalized random variables are uncorrelated.
Todemonstratetheusefulnessofthenormalization,wesolvedamulti-itemcontinuousreviewinventory(Q,r)model
in which the vanilla box and optional components are correlated under stochastic payment and budget constraints.
With normalization, we showed that the set of equations coupled with the vanilla box and optional components was
decoupled into sets of equations for the normalized quantities. Furthermore, each set of decoupled equations was
reduced to a closed form and could be solved numerically without any approximation.
We also performed the sensitivity analysis in terms of the correlation. We found that the order quantity and the
reorder point of optional components depended on the strength of the correlation as we expected. In particular,
we showed that the order quantity of an optional component reached its maximum when there was no correlation
betweenthevanillaboxandtheoptionalcomponent. Inaddition,thereorderpointofanoptionalcomponentreached
10
a maximum that depended on the safety stock of the vanilla box.
Theproposedmethodcanbeusedasadependablemethodforageneralizedmulti-itemcontinuousreviewinventory
modelwithcomplicatedinteractionsamongitems. Itwouldbeinterestingtoinvestigatehowfartheproposedmethod
can be applied to other types of correlation, such as a multivariate Gaussian or other multivariate distributions.
Acknowledgments
This researchwas supported by the Basic Science ResearchProgramthroughthe NationalResearchFoundationof
South Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2010-0022163).
VI. APPENDIX I: SIMPLIFICATION OF VARIOUS FUNCTIONS
In this appendix, we show how to simplify the expressions in Eqs. (15)–(18). To this end, we define various
normalized variables as follows:
x µ r µ
v v v
z − and z − , (A-1)
v
≡ σ ≡ σ
v v
x µ r µ
oj oj oj
z − and z − , (A-2)
j oj
≡ σ ≡ σ
oj oj
where
σ
σ σ 1 ρ2 and µ µ +ρ j(r µ ) . (A-3)
oj ≡ j − j oj ≡ j jσ v− v
v
q
In addition, from Eq. (3), we have
1 (x µ )2 1 (x µ )2
v oj
f (x)= exp − and f (xr )= exp − . (A-4)
Xv √2πσv (cid:26)− 2σv2 (cid:27) Xj|Xv | v √2πσoj (− 2σo2j )
Finally, the G-function is defined as
G(z) ∞ 1 e−z2/2dz . (A-5)
≡ √2π
Zz
A. Evaluation of Rr∞v(x−rv)fXv(x)dx and drdv Rr∞v(x−rv)fXv(x)dx
From Eqs. (A-1) and (A-4), we have
∞(x−rv)fXv(x)dx= √σ2vπ ∞(z−zv)e−z2/2dz =σv √12πe−zv2/2−zvG(zv) . (A-6)
Zrv Zzv (cid:26) (cid:27)
By using the above result, its derivative becomes
ddrv Zrv∞(x−rv)fXv(x)dx=σv ddzrvv ddzv (cid:26)√12πe−zv2/2−zvG(zv)(cid:27)=−G(zv) . (A-7)
B. Evaluation of Rr∞oj(x−roj)fXj|Xv(x|rv)dx and ∂r∂oj Rr∞oj(x−roj)fXj|Xv(x|rv)dx
From Eqs. (A-2)–(A-4), we have
∞ ∞ 1 (x µ )2
oj
(x r )f (xr )dx= (x r ) exp − dx
Zroj − oj Xj|Xv | v Zroj − oj √2πσoj (− 2σo2j )
= σj 1−ρ2j ∞(zj −zoj)√12πe−zj2/2dzj =σoj √12πe−zo2j/2−zojG(zoj) . (A-8)
q Zzoj (cid:26) (cid:27)